Complete simplicial fans, Stanley--Reisner rings, and equivariant h-polynomials

Technical Summary: Complete Simplicial Fans, Stanley–Reisner Rings, and Equivariant h-polynomial

Problem Statement
The paper addresses the combinatorial and algebraic study of the action of finite groups on the cohomology of toric varieties and their associated algebraic structures. Specifically, it focuses on the case where a finite group $G$ acts linearly on a vector space $V$, preserving a complete simplicial fan $\Sigma$. While the cohomology of compact toric orbifolds (associated with rational complete simplicial fans) is well-understood via the Stanley–Reisner ring $A(\Sigma)$ and its Artinian reduction, the graded character of the group action on this ring had not been fully characterized for general fans. Furthermore, the paper investigates the combinatorial structure of the invariants of these rings under the action of finite reflection groups, seeking a combinatorial object that encodes the topology of the quotient space.

Methodology
The author employs a synthesis of commutative algebra, representation theory, and convex geometry:

1. Algebraic Framework: The study utilizes the Stanley–Reisner ring $A(\Sigma)$ of a complete simplicial fan $\Sigma$, defined as a quotient of a polynomial ring by a square-free monomial ideal. The ring is treated as a graded module over a polynomial ring $A = \text{Sym}(V^*)$. The "Artinian reduction" $\overline{A(\Sigma)}$ is obtained by quotienting $A(\Sigma)$ by a linear system of parameters, which corresponds to the usual cohomology of the associated toric variety when $\Sigma$ is rational.
2. Equivariant Character Theory: The paper derives a graded character formula for the action of an element $g \in G$ on $\overline{A(\Sigma)}$. This involves decomposing the ring into subspaces indexed by cones, analyzing the fixed-point sets of the group action on the fan, and utilizing the induced quotient representations $\psi_\sigma: G_\sigma \to \text{GL}(V/V_\sigma)$.
3. Reflection Group Analysis: For the specific case where $G$ is a finite reflection group $W$, the author introduces a new combinatorial object called the "hybrid fan" $\Sigma_W$. This construction, attributed to Colin Crowley and Connor Simpson, encodes the $W$-action by combining rays from the fundamental domain of $W$ with negative simple roots.
4. Induced Representations and Frobenius Reciprocity: The proofs rely heavily on the theory of induced representations, Frobenius reciprocity, and the properties of parabolic subgroups within reflection groups to relate the invariants of the original fan to the combinatorics of the hybrid fan.

Key Contributions and Results

• Equivariant h-polynomial Formula (Theorem 1.1):
  The primary result is a graded character formula for the action of any finite group $G$ on the Artinian reduction $\overline{A(\Sigma)}$. The equivariant Hilbert series $R_{A(\Sigma)}(t)$ is given by:
  $$R_{A(\Sigma)}(t) = \sum_{\sigma \in \Sigma} \frac{|G_\sigma|}{|G|} \text{Ind}_{G_\sigma}^G (\det(tI - \psi_\sigma))$$
  where $G_\sigma$ is the setwise stabilizer of a cone $\sigma$, and $\psi_\sigma$ is the induced representation on the quotient space $V/V_\sigma$. This formula generalizes the classical $h$-polynomial (recovered when $g=\text{id}$) and recovers known results for Weyl groups acting on permutohedra (Procesi, Dolgachev–Lunts, Stembridge, Lehrer).

• Palindromicity (Proposition 3.2):
  The paper proves that the graded character $P_G(t, g)$ is palindromic, satisfying $P_G(t, g) = t^d P_G(t^{-1}, g)$, where $d = \dim(V)$. This is established by showing that the top-degree component of the Artinian reduction carries the trivial representation, a consequence of the Gorenstein property of the Stanley–Reisner ring and the preservation of a volume element under the group action.

• Hybrid Fan and Invariants (Theorem 1.2):
  When $W$ is a finite reflection group, the Poincaré polynomial of the $W$-invariants of $\overline{A(\Sigma)}$ is shown to be equal to the $h$-polynomial of a newly defined "hybrid fan" $\Sigma_W$. The hybrid fan is constructed by taking rays in the fundamental domain and adding rays spanned by negative simple roots, with cone structures determined by the orbits of the original fan.

• Cohomology of Quotients (Theorem 1.3):
  As a direct application, if $\Sigma$ is a rational complete simplicial fan, the Poincaré polynomial of the cohomology of the quotient toric orbifold $X_\Sigma/W$ is equal to the Poincaré polynomial of the toric orbifold $X_{\Sigma_W}$ associated with the hybrid fan. This generalizes previous results by Horiguchi–Masuda–Shareshian–Song and Crowley–Gong–Simpson regarding quotients of polytopes and their normal fans.

Significance and Claims
The paper claims to provide a unified combinatorial framework for understanding the equivariant cohomology of toric varieties under finite group actions, extending beyond the rational case to general complete simplicial fans.

• It establishes that the "equivariant $h$-polynomial" serves as a complete invariant for the graded representation of the group on the cohomology of the associated toric variety (in the rational case).
• It demonstrates that the complex problem of computing the cohomology of a quotient space $X_\Sigma/W$ can be reduced to the combinatorial computation of the $h$-polynomial of the hybrid fan $\Sigma_W$, even when the original fan is non-rational.
• The work bridges the gap between the combinatorics of fans and the topology of quotient spaces, suggesting that the hybrid fan is the correct combinatorial model for the quotient.

Open Questions and Future Directions
The paper concludes by posing several open problems:

1. Ring Isomorphism: Whether an explicit graded ring isomorphism exists between the invariants $\overline{A(\Sigma)}^W$ and the Stanley–Reisner ring of the hybrid fan $\overline{A(\Sigma_W)}$.
2. Variety Isomorphism (Conjecture 5.2): Whether the quotient toric variety $X_\Sigma/W$ is isomorphic to the toric variety $X_{\Sigma_W}$ for any rational fan. The paper notes this is known for polytopal fans and single cones but remains open for general fans.
3. Intersection Cohomology: Whether a similar equivariant character formula exists for the combinatorial intersection cohomology of general (non-simplicial) fans, potentially involving equivariant Kazhdan–Lusztig–Stanley polynomials.
4. Quotient Intersection Cohomology: Whether the invariants of the combinatorial intersection cohomology of a fan equal the intersection cohomology of the hybrid fan, noting that singularities in the quotient may complicate this relationship.